112 XIII INTEGRATION 9. If/is any real-valued function on 1= [0, 1], the nth Bernstein polynomial offis defined to be (a) If /^ 0 on I, then Bn / i> 0 on I, and Bn i is equal to the constant 1 . Deduce that HB.,,11 ^ H/ll in *<I). (b) Show by induction on k that, if fk(t) = /*, we have (**) Bn,/fc(0 = <Wfc + P*,n(0, where akr n = (1 (k l)/ri)ak-ltn and <30>n = 1, and P*,n is a polynomial of degree <^A; 1 whose coefficients are less than Ck/n in absolute value, where C* is a constant independent of n. (Differentiate the equation (*fc) with respect to /, then multiply through by /.) (c) Deduce from (b) and Weierstrass' theorem that, for every continuous function /e ^(1), the sequence (Bn,/) converges uniformly to /on I. 10. Let X be a metrizable compact space, let (/,)ť o be a sequence of complex-valued continuous functions on X, and let (cn)n250 be a sequence of complex numbers. (a) In order that there should exist a complex measure jit on X such that ju(/rt) = cn for all Ť, it is necessary and sufficient that there should exist a number A > 0 with the following property: for every finite sequence (^k)o^k^n of complex numbers, we have (Use the Hahn-Banach theorem.) (b) Suppose that the fk are real-valued, the numbers ck real, and/0 = 1. Then there exists a positive measure ju, on X such that fj.(fn) ~ cn for all n if and only if, for each n sequence (Xk)0^kŁn of real numbers such that J]Xkfk(x) ^0 for all jceX, we have *=o V A* cft > 0 (cf. Section 13.3, Problem 2). fc = 0 11. In Problem 10, take X= [0, 1] and /(/) = /" ("Hausdorff's moment problem"). For each sequence (cn) of scalars, put (a) Show that there exists a complex measure p, on X such that //.(/,) = cn for all n if and only if there exists a number A > 0 such that it) for all n. (Observe that A*cn must be the value of p for the polynomial /"(I t)k, and use Problem 9(b) by remarking that, for each real polynomial P, there exists a constant CP such that Bn,P~-Cp^P^Bn,pH-icP n n for all n.) (b) There exists a positive measure p, on X such that p,(fn) = cn for all n if and only if A*cw ^ 0 for all k ^ 0 and all n ^ 0 (same method).Bohl's theorem")- (Use